A cube of edge 2b is centered at the origin. A very long, straight wire located along the z axis carries a current I in the z direction. Find the flux (4) passing through the surface at x = b. xdx [Hint: £. = – sin ø = Tyi J4g= In|x² + a²|+ C) x2+y2 x2+q2
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- longitudinally-uniform, and axially-symmetric 2. The current distribution of an infinite, wire can be described in cylindrical coordinates by J = J(p)ź. (a) Show that · ƒ = 0. (b) Considerations of longitudinal and axial symmetry require that the mag- netic field can only depend upon p, i.e. that = Ẻ(p) = B₂(p)ô+ Bø(p)❖ + B₂(p)ź. Use Ampère's Law to determine Bo(p) in terms of I(p) = 2π ff J(p') p'dp'. (c) Use the Biot and Savart Law to show that Bp(p) = B₂ (p) = 0. Side note: it is pretty easy to show that Bp(p) must be zero using Gauss's Law for B with a cylindrical volume. I am not aware of an "easy" way to see B(p) 0 not that it is very difficult using the Biot and Savart Law... (d) Use the integral form for the vector potential [Jackson, Eq. (5.32)] to determine A for this current distribution. Hint: in order to deal with divergent integrals, you may want to limit the current distribution to -L≤ zThe unit of surface current is A/m^2. Select one: True O False5. Consider two (1 and 2) wires both having constant circular cross sectional areas: wire 1 has diameter = d, ; wire 2 has diameter = d. Potential differences are applied across each wire so that they both carry current. The current in wire 1 is I, and in wire 2 it is I, ; the the magnitude of the current density in wire 1 is J, and in wire 2 it is J,. If I,= I, and J, = (2) J, then wire diameters must be such that d, = (?) d. a. 2/2 b. V212 c. 2 d. V2 e. V214Two long thin conducting tubes are concentric with each other. The outer tube has a radius R and has a current I while the inner tube has a radius R/4. To cancel the field at points r > R, the inner tube must have a current a.) running parallel to I of magnitude I/4 b.) running parallel to I of magnitude I c.) running antiparallel to I of magnitude 1/4 d.) running antiparallel to I of magnitude IThe figure shows a cross section of a long, conducting coaxial cable of radii a, b and c. Uniform current density flows through the two conductors such that the total current through the inner conductor is I and total current through the outer conductor is I. Derive the expression for B(r) in the regions ra and sketch a plot of B(r).A cube has two rings on opposite faces of the cube. The rings carry equal currents i. The rings are sized such that their diameters are equal the side length of the cube s. What is the field strength in the center of the cube if the currents run such that the field strength is as big as possible? Start with a derivation of B from a ring. Please include a diagram for the problem.A metallic sheet lies on the xy plane, carries a surface current density K = Ho where μo is the permeability of free space. The B-field just below the sheet is given as B = 2î + 4ĵ + 5k. The questions on this page are based on this scenario. -Ĵ, What is the z-component of the B-field just above the sheet?Figure 8 shows a coaxial cable (two nested cylinders) of length l, inner radius a and outer radius b. Note that l >> a and l >> b. The inner cylinder is charged to +Q and the outer cylinder is charged to −Q. The cable carries a current I, which flows clockwise. Use Ampere’s Law to calculate themagnetic field B⃗ at r<a, a<r<b, and r>b.A square loop has a charge density λ = |x| at y = ±a and [x]